Optimal. Leaf size=125 \[ \frac {e \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^7 (b d-a e)}{4 b^3}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^6 (b d-a e)^2}{7 b^3}+\frac {e^2 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^8}{9 b^3} \]
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Rubi [A] time = 0.18, antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {770, 21, 43} \begin {gather*} \frac {e \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^7 (b d-a e)}{4 b^3}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^6 (b d-a e)^2}{7 b^3}+\frac {e^2 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^8}{9 b^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 21
Rule 43
Rule 770
Rubi steps
\begin {align*} \int (a+b x) (d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int (a+b x) \left (a b+b^2 x\right )^5 (d+e x)^2 \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac {\left (b \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int (a+b x)^6 (d+e x)^2 \, dx}{a b+b^2 x}\\ &=\frac {\left (b \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int \left (\frac {(b d-a e)^2 (a+b x)^6}{b^2}+\frac {2 e (b d-a e) (a+b x)^7}{b^2}+\frac {e^2 (a+b x)^8}{b^2}\right ) \, dx}{a b+b^2 x}\\ &=\frac {(b d-a e)^2 (a+b x)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{7 b^3}+\frac {e (b d-a e) (a+b x)^7 \sqrt {a^2+2 a b x+b^2 x^2}}{4 b^3}+\frac {e^2 (a+b x)^8 \sqrt {a^2+2 a b x+b^2 x^2}}{9 b^3}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 217, normalized size = 1.74 \begin {gather*} \frac {x \sqrt {(a+b x)^2} \left (84 a^6 \left (3 d^2+3 d e x+e^2 x^2\right )+126 a^5 b x \left (6 d^2+8 d e x+3 e^2 x^2\right )+126 a^4 b^2 x^2 \left (10 d^2+15 d e x+6 e^2 x^2\right )+84 a^3 b^3 x^3 \left (15 d^2+24 d e x+10 e^2 x^2\right )+36 a^2 b^4 x^4 \left (21 d^2+35 d e x+15 e^2 x^2\right )+9 a b^5 x^5 \left (28 d^2+48 d e x+21 e^2 x^2\right )+b^6 x^6 \left (36 d^2+63 d e x+28 e^2 x^2\right )\right )}{252 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 1.90, size = 0, normalized size = 0.00 \begin {gather*} \int (a+b x) (d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.41, size = 234, normalized size = 1.87 \begin {gather*} \frac {1}{9} \, b^{6} e^{2} x^{9} + a^{6} d^{2} x + \frac {1}{4} \, {\left (b^{6} d e + 3 \, a b^{5} e^{2}\right )} x^{8} + \frac {1}{7} \, {\left (b^{6} d^{2} + 12 \, a b^{5} d e + 15 \, a^{2} b^{4} e^{2}\right )} x^{7} + \frac {1}{3} \, {\left (3 \, a b^{5} d^{2} + 15 \, a^{2} b^{4} d e + 10 \, a^{3} b^{3} e^{2}\right )} x^{6} + {\left (3 \, a^{2} b^{4} d^{2} + 8 \, a^{3} b^{3} d e + 3 \, a^{4} b^{2} e^{2}\right )} x^{5} + \frac {1}{2} \, {\left (10 \, a^{3} b^{3} d^{2} + 15 \, a^{4} b^{2} d e + 3 \, a^{5} b e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (15 \, a^{4} b^{2} d^{2} + 12 \, a^{5} b d e + a^{6} e^{2}\right )} x^{3} + {\left (3 \, a^{5} b d^{2} + a^{6} d e\right )} x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.18, size = 379, normalized size = 3.03 \begin {gather*} \frac {1}{9} \, b^{6} x^{9} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{4} \, b^{6} d x^{8} e \mathrm {sgn}\left (b x + a\right ) + \frac {1}{7} \, b^{6} d^{2} x^{7} \mathrm {sgn}\left (b x + a\right ) + \frac {3}{4} \, a b^{5} x^{8} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {12}{7} \, a b^{5} d x^{7} e \mathrm {sgn}\left (b x + a\right ) + a b^{5} d^{2} x^{6} \mathrm {sgn}\left (b x + a\right ) + \frac {15}{7} \, a^{2} b^{4} x^{7} e^{2} \mathrm {sgn}\left (b x + a\right ) + 5 \, a^{2} b^{4} d x^{6} e \mathrm {sgn}\left (b x + a\right ) + 3 \, a^{2} b^{4} d^{2} x^{5} \mathrm {sgn}\left (b x + a\right ) + \frac {10}{3} \, a^{3} b^{3} x^{6} e^{2} \mathrm {sgn}\left (b x + a\right ) + 8 \, a^{3} b^{3} d x^{5} e \mathrm {sgn}\left (b x + a\right ) + 5 \, a^{3} b^{3} d^{2} x^{4} \mathrm {sgn}\left (b x + a\right ) + 3 \, a^{4} b^{2} x^{5} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {15}{2} \, a^{4} b^{2} d x^{4} e \mathrm {sgn}\left (b x + a\right ) + 5 \, a^{4} b^{2} d^{2} x^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {3}{2} \, a^{5} b x^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) + 4 \, a^{5} b d x^{3} e \mathrm {sgn}\left (b x + a\right ) + 3 \, a^{5} b d^{2} x^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{3} \, a^{6} x^{3} e^{2} \mathrm {sgn}\left (b x + a\right ) + a^{6} d x^{2} e \mathrm {sgn}\left (b x + a\right ) + a^{6} d^{2} x \mathrm {sgn}\left (b x + a\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 271, normalized size = 2.17 \begin {gather*} \frac {\left (28 b^{6} e^{2} x^{8}+189 x^{7} e^{2} a \,b^{5}+63 x^{7} d e \,b^{6}+540 x^{6} e^{2} a^{2} b^{4}+432 x^{6} d e a \,b^{5}+36 x^{6} d^{2} b^{6}+840 x^{5} e^{2} a^{3} b^{3}+1260 x^{5} d e \,a^{2} b^{4}+252 x^{5} d^{2} a \,b^{5}+756 a^{4} b^{2} e^{2} x^{4}+2016 a^{3} b^{3} d e \,x^{4}+756 a^{2} b^{4} d^{2} x^{4}+378 x^{3} e^{2} a^{5} b +1890 x^{3} d e \,a^{4} b^{2}+1260 x^{3} d^{2} a^{3} b^{3}+84 x^{2} e^{2} a^{6}+1008 x^{2} d e \,a^{5} b +1260 x^{2} d^{2} a^{4} b^{2}+252 a^{6} d e x +756 a^{5} b \,d^{2} x +252 d^{2} a^{6}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}} x}{252 \left (b x +a \right )^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.76, size = 452, normalized size = 3.62 \begin {gather*} \frac {1}{6} \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a d^{2} x - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{3} e^{2} x}{6 \, b^{2}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} e^{2} x^{2}}{9 \, b} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{2} d^{2}}{6 \, b} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{4} e^{2}}{6 \, b^{3}} - \frac {11 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} a e^{2} x}{72 \, b^{2}} + \frac {83 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} a^{2} e^{2}}{504 \, b^{3}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} {\left (2 \, b d e + a e^{2}\right )} a^{2} x}{6 \, b^{2}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} {\left (b d^{2} + 2 \, a d e\right )} a x}{6 \, b} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} {\left (2 \, b d e + a e^{2}\right )} a^{3}}{6 \, b^{3}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} {\left (b d^{2} + 2 \, a d e\right )} a^{2}}{6 \, b^{2}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} {\left (2 \, b d e + a e^{2}\right )} x}{8 \, b^{2}} - \frac {9 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} {\left (2 \, b d e + a e^{2}\right )} a}{56 \, b^{3}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} {\left (b d^{2} + 2 \, a d e\right )}}{7 \, b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \left (a+b\,x\right )\,{\left (d+e\,x\right )}^2\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a + b x\right ) \left (d + e x\right )^{2} \left (\left (a + b x\right )^{2}\right )^{\frac {5}{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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